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I am pretty much through with expanding and polishing
Now I am starting to work on
So far I have restructured the section outline at these entries to match that of the entry cohesive infinity-groupoid. All these entries are supposed to go through that list of Structures in a cohesive $\infty$-topos in parallel.
I have also removed at smooth infinity-groupoid a bunch of material that is meanwhile discussed in more polished form at Euclidean-topological infinity-groupoid.
To account for that I should now add a discussion of a free-forgtful adjunction between $Smooth \infty Grpd$ and $ETop \infty Grpd$ and how to transfer structure along that.
I’ve been reworking smooth infinity-groupoid.
I have reorganized all the material. Instead of the previous list of examples, which discussed all kinds of structure on a given class of smooth $\infty$-groupoids, I have instead now a list of all the structures, and subsections with realizations on specific classes of smooth $\infty$-groupoids.
Then I moved over from circle n-bundle with connection all material on intrinsic de Rham cohomology and intrinsic differential cohomology that had not already been there. I have come to think that it is best to discuss all the intrinsic structure of a cohesive $\infty$-topos in one place, not splitting them off in separate entries.
I started polishing mostly the subsections that deal with the circle $n$-groups $\mathbf{B}^n U(1)$. But not done yet.
I have added to smooth infinity-groupoid in the subsection Concrete objects the observation that every diffeological Kan complex identifies as an object in $Smooth \infty Grpd$ that is a concrete object in the sense of cohesive oo-toposes the way I have currently defined it there.
But I am clearly still missing the central statements about concrete objects. Whatever they are.
I have started now working on the section Exponentiated oo-Lie algebras.
So far I have mostly removed material that had meanwhile found its way in my article with Domenico and Jim in improved form and which I need to re-insert here after reorganizing a bit.
Then I have added the proof that for $\mathfrak{g}$ any $L_\infty$-algebra, the smooth $\infty$-groupoid $\exp(\mathfrak{g})$ (its “$\infty$-simply connected Lie integration”) is geometrically contractible, in that
$\Pi \exp(\mathfrak{g}) \simeq * \,,$thus quallifying as an exponentiated $\infty$-Lie algebra also in the general abstract sense of cohesive $\infty$-toposes.
The contractibility statement is an easy observation in André’s original article in the context of simplicial Banach spaces. The point of the proof here is to observe that everything goes through in the diffeological $\infty$-groupoid case, too, and that the “naive” contractibility is indeed that as seen by the intrinsic fundamental $\infty$-groupoid functor $\Pi$. I tried to write that out carefully, which maybe makes the proof unduly lengthy, but anyway.
have added the observation that for $\mathfrak{g}$ any $L_\infty$-algebra, the smooth $\infty$-groupoid $\exp(\mathfrak{g})$ is a concrete object (here).
I have fully formalized the proposition that for $G \in SmoothMfd \hookrightarrow Smooth \infty Grpd$ a Lie group, the canonical morphism $\theta : G \to \mathbf{\flat}_{dR}\mathbf{B}G$ identifies with the ordinary Maurer-Cartan form of $G$.
(This statement itself is in the section The canonical form on a Lie group and relies on previous subsections being linked to from there.)
I have worked on the section Exponentiated oo-Lie algebras.
The materal that the section does contain is (hopefully) somewhat polished and comprehensive now, but there is more material that could/should eventually go here.
I have been working on the sections Exponentiated oo-Lie algebra – flat coefficients and Exponentiated oo-Lie algebra – de Rham coefficients bringing the discussion of the various presentations of the differential coefficient objcts for the line $n$-group into shape.
Using this I have then included the discussion of the universal curvature characteristic on $\mathbf{B}^n U(1)$ in terms of these presentations at Universal curvature characteristic on circle n-group .
This culminates in the proof that the universal curvature form is presented by the left top vertical morphism in the double pullback of simplicial presheaves of differential forms whose components over $U \in CartSp$ in degree $[k] \in \Delta$ are indicated here (just for the fun of it):
$\array{ \left\{ \array{ \Omega^\bullet_{si, vert}(U \times \Delta^k) &\stackrel{A_{vert}}{\leftarrow}& CE(b^{n-1}\mathbb{R}) \\ \uparrow && \uparrow \\ \Omega^\bullet_{si}(U \times \Delta^k) &\stackrel{A}{\leftarrow}& W(b^{n-1}\mathbb{R}) } \right\} &\to& \left\{ \array{ \Omega^\bullet_{si, vert}(U \times \Delta^k) &\stackrel{}{\leftarrow}& W(b^{n-1}\mathbb{R}) \\ \uparrow && \uparrow^{\mathrlap{id}} \\ \Omega^\bullet_{si}(U \times \Delta^k) &\stackrel{}{\leftarrow}& W(b^{n-1} \mathbb{R}) } \right\} \\ \downarrow && \downarrow \\ \left\{ \array{ \Omega^\bullet_{si, vert}(U \times \Delta^k) &\stackrel{0}{\leftarrow}& 0 \\ \uparrow && \uparrow \\ \Omega^\bullet_{si}(U \times \Delta^k) &\stackrel{}{\leftarrow}& CE(b^{n} \mathbb{R}) } \right\} &\to& \left\{ \array{ \Omega^\bullet_{si,vert}(U \times \Delta^k) &\leftarrow& CE(b^{n} \mathbb{R}) \\ \uparrow && \uparrow^{\mathrlap{id}} \\ \Omega^\bullet_{si}(U \times \Delta^k) &\stackrel{}{\leftarrow}& CE(b^{n} \mathbb{R}) } \right\} \\ \downarrow && \downarrow \\ * &\to& \left\{ \array{ \Omega^\bullet_{si,vert}(U \times \Delta^k) &\leftarrow& CE(b^{n} \mathbb{R}) \\ \uparrow && \uparrow \\ \Omega^\bullet_{si}(U \times \Delta^k) &\stackrel{}{\leftarrow}& 0 } \right\} }$I think I am pretty much through with the section Chern-Weil homomorphism at smooth infinity-groupoid. I need to add a concluding paragraph to that section, but otherwise I think I will continue further in-depth discussion at the corresponding dedicated entries connection on an infinity-bundle and infinity-Chern-Weil theory (which are badly in need of some polishing,, each).
So the idea would be that the general abstract structures whose realization in $Smooth \infty Grpd$ I did unwind to this point serve as the basic software library/API for implementations in $Smooth \infty Grpd$ that connect the general abstract templates to some standard routines, but that all further special purpose applications be discussed on their separate pages.
At smooth infinity-groupoid I have made explicit the essential geometric morphism $i : Smooth \infty Grpd \to ETop \infty Grpd$ in a new section Relative cohesion.
Then in the section on Geometric homotopy and Galois theory I have used this to give a more systematic and more comprehensive discussion of the fundamental $\infty$-groupoid functor $\Pi_{Smooth}$ as the composite $\Pi_{ETop} \circ i_!$.
At the heart of this is something like a lengthy lemma about essential $\infty$-geometric morphisms induced by certain morphisms of sites. The details for that I have spelled out and polished at cohesive infinity-groupoid in the section Infinitesimal cohesion – Properties. But eventually this should be discussed elsewhere in full generality.
With Thomas Nikolaus and Danny Stevenson we are currently working on writing out more explicit models of principal $\infty$-bundles by smooth simplicial principal bundles.
As a first little result we have the computation of the canonical form on a smooth $\infty$-group $\theta : G \to \mathbf{\flat}_{dR} \mathbf{B}G$ worked out for the general case that $G$ is presented by a simplicial Lie group.
The result I have written up now in the new section The canonical form on a simplicial Lie group .
The result is what one would hope and expect to see: it is simply a 1-form with values in the simplicial Lie algebra of $G$ which is degreewise the ordinary Maurer-Cartan form. But it is nice to see it drop out from the general machinery.
I have added pointer to
to various related entries, such as smooth infinity-groupoid, shape modality, Haefliger groupoid.
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